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Theorem psubssat 34555
Description: A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
Hypotheses
Ref Expression
atpsub.a 𝐴 = (Atoms‘𝐾)
atpsub.s 𝑆 = (PSubSp‘𝐾)
Assertion
Ref Expression
psubssat ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)

Proof of Theorem psubssat
Dummy variables 𝑞 𝑝 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . 3 (le‘𝐾) = (le‘𝐾)
2 eqid 2621 . . 3 (join‘𝐾) = (join‘𝐾)
3 atpsub.a . . 3 𝐴 = (Atoms‘𝐾)
4 atpsub.s . . 3 𝑆 = (PSubSp‘𝐾)
51, 2, 3, 4ispsubsp 34546 . 2 (𝐾𝐵 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟(le‘𝐾)(𝑝(join‘𝐾)𝑞) → 𝑟𝑋))))
65simprbda 652 1 ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  wss 3559   class class class wbr 4618  cfv 5852  (class class class)co 6610  lecple 15880  joincjn 16876  Atomscatm 34065  PSubSpcpsubsp 34297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-psubsp 34304
This theorem is referenced by:  psubatN  34556  paddidm  34642  paddclN  34643  paddss  34646  pmodlem1  34647  pmod1i  34649  pmodl42N  34652  elpcliN  34694  pclidN  34697  pclbtwnN  34698  pclunN  34699  pclun2N  34700  pclfinN  34701  polssatN  34709  psubclsubN  34741
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